Problem: You have found the following ages (in years) of all 6 sloths at your local zoo: $ 2,\enspace 3,\enspace 1,\enspace 1,\enspace 5,\enspace 7$ What is the average age of the sloths at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Solution: Because we have data for all 6 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $6$ ages and divide by $6$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6}} $ $ {\mu} = \dfrac{2 + 3 + 1 + 1 + 5 + 7}{{6}} = {3.2\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $2$ years $-1.2$ years $1.44$ years $^2$ $3$ years $-0.2$ years $0.04$ years $^2$ $1$ year $-2.2$ years $4.84$ years $^2$ $1$ year $-2.2$ years $4.84$ years $^2$ $5$ years $1.8$ years $3.24$ years $^2$ $7$ years $3.8$ years $14.44$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{1.44} + {0.04} + {4.84} + {4.84} + {3.24} + {14.44}} {{6}} $ $ {\sigma^2} = \dfrac{{28.84}}{{6}} = {4.81\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{4.81\text{ years}^2}} = {2.2\text{ years}} $ The average sloth at the zoo is 3.2 years old. There is a standard deviation of 2.2 years.